A predictive model for rare earth element partitioning between ...

Contrib Mineral Petrol (1997) 129: 166±181

Ó Springer-Verlag 1997

Bernard J. Wood
Jonathan D. Blundy

A predictive model for rare earth element partitioning between clinopyroxene and anhydrous silicate melt

Received: 8 November 1996 / Accepted: 28 May 1997

Abstract We present a quantitative model to describe the partitioning of rare earth elements (REE) and Y between clinopyroxene and anhydrous silicate melt as a function of pressure …P †, temperature …T † and bulk composition …X †. The model is based on the Brice (1975) equation, which relates the partition coecient of element i …Di † to that of element o …Do † where the latter has the same ionic radius ro as the crystallographic site of interest, in this case the clinopyroxene M2 site: 0 Di ˆ Do exp@

ÿ4pEM2 NA



ro 2

…ri ÿ ro †2 ‡ 13 …ri ÿ ro †3 RT

1 A

NA is Avogadro's number, EM2 is the Young's Modulus of the site, R is the gas constant and T is in K. Values of EM2 obtained by ®tting the Brice equation to experimental REE partition coecient patterns are in good agreement with those obtained from the well-known correlation between bulk modulus, metal-oxygen distance and cation charge. Using this relationship to constrain EM2 for 3+ cations and then ®tting the Brice equation to those experimental data where 3 or more REE partition coecients had been simultaneously measured we obtained 82 values of Do and ro . The latter was found to be a simple and crystallochemically reasonable function of clinopyroxene composition. We show that for any clinopyroxene-melt pair if D for one middle REE (e.g. Sm or Gd) is known then the Brice equation can be used to predict Ds for all the other REE, with uncertainties similar to those involved in the actual measurements. The model was generalised using thermodynamic descriptions of REE components in crystal and melt phases to estimate the free energy of fusion …DGf † of the ®ctive REE components REEMgAlSiO6

B.J. Wood (&)
J.D. Blundy CETSEI, Department of Geology, University of Bristol, Bristol, BS8 1RJ, UK Editorial responsibility: A. Hofman

and Na0:5 REE0:5 MgSi2 O6 . For the melt we ®nd that 6-oxygen melt components (CaMgSi2 O6 ; NaAlSi2 O6 ; Mg3 Si1:5 O6 etc.) mix with constant activity coecient over a wide range of natural compositions. Propagating DGf into the Brice model we obtain an expression for D3‡ o in terms of the atomic fraction of Mg on the clinopyroxene M1 site, the Mg-number of the melt, P and T . The D for any REE can be calculated from D3‡ o using the Brice equation. Over 92% of DREE (454 points) calculated in this way lie within a factor 0.63±1.59 of the experimental value. The approach can be extended to calculate D for any REE at a given P … 6 GPa† and T …1200±2038 K† to within 0.60±1.66 times the true value given only the crystal and/or melt composition. The model has widespread applicability to geochemical modelling of all natural processes involving clinopyroxene, e.g. decompression mantle melting, enabling for the ®rst time account to be taken of variations in partition coecient in response to changing pressure, temperature and phase composition.

Introduction Since the development of accurate methods of determining element concentration at the parts per million (ppm) level the trace element contents of igneous rocks have frequently been used to model their chemical evolution. These studies use estimated crystal-liquid partition coecients together with solutions for the di€erential equations describing, for example, fractional crystallisation or fractional melting (Shaw 1970) to model evolution of the melt during precipitation or dissolution of the crystalline phases. Generally, because of lack of data, the crystal-liquid partition coecients are assumed to be constant during di€erentiation, an assumption which is thermodynamically implausible and which contradicts observed variations in partition coef®cients (Jones 1995). This is a fundamental limitation in the use of trace element partition coecients in petrogenetic modelling of polythermal, polybaric processes.

167

Until the e€ects of pressure, temperature and composition on element partitioning are known, the results, independent of the type of fractionation process involved, are bound to be subject to large uncertainty. Our aim here is to develop a method of predicting variations in partition coecients using thermodynamic principles in combination with a recently developed model which relates partitioning to ionic radius, lattice strain and elastic constants. Because of their petrogenetic importance we focus mainly on the partitioning of rare earth elements (REE) between clinopyroxene and melt. However the approach can be generalised to other elements and minerals.

Background Consideration of crystal-liquid equilibrium from a thermodynamic standpoint indicates that partitioning should depend on pressure, temperature and bulk composition (e.g. Wood and Fraser 1976, pp. 199±200). Separating the di€erent e€ects is dicult, however, because, for any given crystal-melt pair, temperature, pressure and composition are strongly correlated. This is illustrated by the partitioning of Sr and Ba between plagioclase feldspar and silicate melt. The partition coecients are apparent functions of temperature and bulk composition and were ®tted by Nielsen (1988) to an equation in reciprocal temperature after taking explicit account of bulk compositional e€ects in the melt using a 2-lattice model. Bulk compositional controls may, however be exercised by the silicate melts or by the plagioclase crystals or by both. Since the plagioclase and melt compositions are strongly correlated, it is dicult to determine their relative importance. Blundy and Wood (1991) argued the opposite case to that assumed by Nielsen (1988), namely that the bulk compositional e€ects of the crystals should dominate over those of the melt. The rationale is that the large cations, Ba and Sr, much more easily replace smaller Na and Ca in the relatively ¯exible melt structure than in the rigid crystal structure. The introduction of large Ba or Sr ions into a smaller lattice site generates elastic strain energy, the amount of which depends on the Young's modulus of the medium. Since the Young's modulus of a phase with no shear modulus, such as silicate melt, is zero, the strain energy must be dominated by the elastic properties of the crystals, which have Young's moduli in the range 100±300 GPa. Blundy and Wood (1991) were able to demonstrate that crystal composition, rather than melt composition or temperature exerts the greatest control on Ba and Sr feldspar-liquid partition coecients, which range from about 0.1 to 8.0 and 1.2 to 33 respectively in igneous systems, by considering the data of Lagache and Dujon (1987). The latter authors found, at constant temperature, a strong dependence of Sr partitioning between plagioclase and aqueous solutions on the anorthite content of the feldspar. Since the solutions they used

were extremely dilute (1 Molal), the e€ects they observed must be almost entirely due to the compositional variations of the crystals. Blundy and Wood (1991) demonstrated that when the plagioclase/¯uid partitioning results were recast on a molar rather than weight basis, the dependence of DSr on XAn is exactly the same as that observed in polythermal plagioclase-silicate melt experiments (Fig. 1). Therefore, solution or melt composition must play a subordinate role to that of the crystal, at least in the case of Sr and Ba partitioning between plagioclase and melt or ¯uid. In order to quantify the e€ects of elastic strain energy on partition coecients, Blundy and Wood (1994) adopted the model of Brice (1975) who developed equations describing the strain energy stored in an isotropic crystal lattice when a cation of radius ri substitutes for the host cation of radius ro . It was shown that partitioning of 1+, 2+ and 3+ cations between plagioclase and melt and between clinopyroxene and melt could quantitatively be described in terms of these strain energies of substitution. Beattie (1994), applying the theoretical model of Nagasawa (1966), arrived at a similar conclusion for trace element partitioning between olivine and melt. In the case of plagioclase, Blundy and Wood (1994) found that the variations of strain energy with charge bear a simple and predictable relationship to the bulk elastic properties of end-member anorthite and albite. In order to develop a predictive model for partitioning of REE between clinopyroxene and silicate melt, we need to join a thermodynamic description of crystal and liquid properties to the strain energy model of Brice (1975) and Blundy and Wood (1994), since the latter applies only at ®xed temperature, pressure and bulk

Fig. 1 Plot, taken from Blundy and Wood (1991), of RT ln DSr for plagioclase-melt partitioning as a function of mole fraction anorthite …XAn † in the crystal. Note that Sr is more compatible in albite …RT ln DSr > 0† than in anorthite …RT ln DSr < 0†. Note also that the isothermal …750  C† plagioclase-hydrothermal ¯uid partitioning data of Lagache and Dujon (1987) (solid circles) are in excellent agreement with the polythermal plagioclase-melt data. Sources of volcanic and experimental data given in Blundy and Wood (1991), except J.D. Blundy (unpublished) and Drake (1972)

168

(major element) composition. In order to maintain a general approach, it is necessary to consider both homovalent (e.g. Sr2‡ for Ca2‡ ) and heterovalent (Na‡ or REE3‡ for Ca2‡ ) substitution. We begin with the case of homovalent substitution.

Theoretical development Consider a hypothetical C2/c clinopyroxene J 2‡ MgSi2 O6 where J 2‡ is a cation of radius ro which ®ts exactly into the large M2 site without straining the structure. The partitioning of J between crystal and silicate melt should then be determined principally by the free energy of the fusion reaction: J MgSi2 O6 ˆ J MgSi2 O6 clinopyroxene melt

…1†

The equilibrium constant Ko is given by the simple relationships: DGoJ ˆ GoJ MgSi2 O6 melt ÿ GoJ MgSi2 O6 cpx Ko ˆ

amelt J MgSi2 O6 acpx J MgSi2 O6

 ˆ exp

ÿDGoJ RT

 …2†

melt where acpx J MgSi2 O6 and aJ MgSi2 O6 refer to the activity of the J MgSi2 O6 component in the clinopyroxene and melt phases respectively. Given activity-composition models for melt and pyroxene, it should be possible to relate Ko to the measured Nernst partition coecient Do , de®ned as:

‰J Šcpx Do ˆ ‰J Šmelt

…3†

where ‰J Šcpx and ‰J Šmelt refer to weight fractions of J 2‡ in clinopyroxene and melt respectively. Before attempting directly to connect Do to Ko let us consider the e€ect of replacing J 2‡ in crystal and melt by very small amounts of cation I 2‡ of the same charge, but di€erent radius ri . Equilibrium of I 2‡ between melt and crystal can be represented in an analogous way to that outlined for J MgSi2 O6 above: IMgSi2 O6 ˆ IMgSi2 O6 clinopyroxene melt

…4†

In this case, however, the IMgSi2 O6 component is at in®nite dilution in a host of essentially pure J MgSi2 O6 . We will assume that if I and J had exactly the same ionic radius then the standard free energy change of reactions (1) and (4) would be the same. The actual di€erence between the standard free energy changes is assumed to be due to the work done in straining crystal and melt by introducing a cation which is not the same size as the site. This is a reasonable assumption for closed-shell ions such as Ca2‡ , Sr2‡ and Mg2‡ and it also appears to work in those cases, such as the lanthanides, where crystal ®eld e€ects are small (Blundy and Wood 1994). Given this assumption the standard free energy change of reaction (4) DGoI , with I in®nitely dilute in both phases, is related to that of reaction (1) by:

cpx DGoI ˆ DGoJ ‡ DGmelt strain ÿ DGstrain

…5†

DGcpx strain

In equation (5), DGmelt refer to the strain strain and energies produced by replacing one mole of J with one mole of I in an in®nite volume of pure J MgSi2 O6 melt and clinopyroxene respectively. If we assume, following Blundy and Wood (1994), that the strain energy term for the melt is negligible with respect to that for the crystal, then we ®nd the following relationship between the equilibrium constants for reactions (1), Ko , and (4), Ki : Ki ˆ

amelt IMgSi2 O6 acpx IMgSi2 O6

ˆ exp

   cpx  ÿDGoJ ‡ DGcpx DGstrain strain ˆ Ko exp RT RT

…6†

Equation (6) shows how the free energy relationships for trace element melting reactions should be related to those for the host cation, given our assumption that strain energy terms dominate. In order to relate the equilibrium constant Ki to partition coecient Di for trace element I we take the activity coecient for J MgSi2 O6 in the crystal to be a constant near 1.0 (Raoult's Law region) and assume that the activity coecients for IMgSi2 O6 and JMgSi2 O6 in the melt are identical. This gives a proportionality between Do and Ko that depends on cmelt J MgSi2 O6 , the activity coecient of J MgSi2 O6 in the melt, and a correction factor for mean molecular weights of melt and crystal: Do ˆ

cmelt J MgSi2 O6
fmeltg

…7†

Ko
fcrystalg

where fmeltg and fcrystalg refer to the mean molecular weights of melt and crystalline phases respectively on a 6-oxygen basis. In a later section we introduce activitycomposition models for the melt, wherein the terms melt cmelt IMgSi2 O6 and cJ MgSi2 O6 approach unity and can therefore be neglected (see below). Given these assumptions, the relationship between Ko and Ki (Eq. 6) can be converted into a relationship between Do and Di as follows:   ÿDGcpx strain Di ˆ Do exp RT

…8†

For a spherical site the strain-free energy term depends (Brice 1975) on the site radius ro , the radius of the substituting ion I, ri , and on the Young's Modulus of the site, ES . Although the clinopyroxene M2 site is far from spherical, having 4 oxygens at 2.36 AÊ distance from the cation, 2 at 2.56 AÊ and 2 at 2.72 AÊ (Smyth and Bish 1988) we will follow Blundy and Wood (1994) and assume that ro is an ``equivalent'' spherical radius which corresponds to that of strain-free substitution. In that case the strain energy term is given by (Brice 1975): DGcpx strain ˆ 4pEM2 NA



ro 1 …ri ÿ ro †2 ‡ …ri ÿ ro †3 3 2



…9†

where NA is Avogadro's Number and EM2 is the Young's Modulus of the M2 site. Combining Eqs. (8) and (9) we obtain an approximately parabolic dependence of Di on ionic radius, with the maximum value of D corresponding to Do , the partition coecient for strain-free substitution in the pyroxene M2 site:

169

is in good agreement with the site radius, and the estimated value of the Young's Modulus of the site ES , approximates that of the bulk Ca-end-member crystal, anorthite or diopside. It is clear, therefore, that homovalent trace element substitution for Ca2‡ in pyroxene and plagioclase follows the Brice Eq. (10) and that knowledge of the crystal-melt partitioning of, for example Ca, enables partitioning of the other 2+ ions to be predicted accurately. The height of the peak, Do depends on Ko which means that it must depend on pressure and temperature in a predictable manner. Before addressing Do in more detail we will consider estimation of ES and ro for the case of heterovalent substitution. Fig. 2 A schematic showing the e€ects on partition coecient of the parameters in the Brice Eq. (10). The ``best-®t'' ion has radius ro and a corresponding partition coecient Do at the peak of the parabola. Partition coecients decrease as ri deviates in either positive or negative direction away from ro . The derivative @D=@ri (or the ``tightness'' of the parabola) increases with increasing ES , the Young's modulus of the site. Do , and to a much lesser extent ES , vary as a function of pressure and temperature; ro varies primarily with crystal composition 0 Di ˆ Do exp@

ÿ4pEM2 NA



ro 2

…ri ÿ ro †2 ‡ 13 …ri ÿ ro †3 RT

1 A

…10†

This theoretical dependence of Di on ionic radius, shown in Fig. 2, corresponds quite closely with observed partition coecient values for 2+ cation substitution in both plagioclase and clinopyroxene (Blundy and Wood 1994). In each case (Figs. 3 and 4a) the peak of the curve

Fig. 3 Brice model ®ts to 1+, 2+ and 3+ cation partitioning between plagioclase …An89 † and silicate melt in the system diopsidealbite-anorthite (Blundy and Wood 1994). Note that the best-®t curves become tighter as the charge on the cation increases from 1+ through 3+. This is due to increase in e€ective Young's Modulus of the site with increasing charge (see text)

Determination of E and ro Rare earth elements in the 3+ oxidation state have similar ionic radii to Ca2‡ and are found to substitute much more readily into Ca-rich pyroxenes than into Capoor pyroxenes (Jones 1995), indicating that they replace Ca2‡ in the M2 site. Given that REE enter the M2 site, one would anticipate that their crystal-liquid partition coecients should, like the 2+ ions, depend on the elastic properties of the crystal. For plagioclase, Blundy and Wood (1994) found that the apparent value of Es2‡ , derived by ®tting Eq. (10) to partitioning data (Fig. 3) is twice the best-®t value of Es1‡ and approximately 0.6 times the value of Es3‡ . Furthermore, the best®t values of Es1‡ and Es2‡ correspond closely, throughout the plagioclase series, to those of pure albite and pure anorthite respectively. Experiments on clinopyroxene-liquid partitioning performed by Blundy and Wood (1994) and Hauri et al. (1994) show a dependence of 1+, 2+ and 3+ cation partitioning on ionic radius and charge (Fig. 4) which is similar to that observed in the plagioclase series. Some deviation from the Brice curve, also noted by Blundy and Wood (1994), can be seen in the cases of nonspherical ions such as Pb2‡ , with its lone pair of electrons, and in transition elements such as Mn and Ni where electronic e€ects may be important. In general, however, closed-shell ions and lanthanides partition in accordance with the Brice model. As can be seen from Fig. 4, the D-ri curves become tighter as cation charge increases from 1+ to 3+ demonstrating, as for plagio1‡ 2‡ 3‡ clase, that EM2 < EM2 < EM2 . The exact relationship is more dicult to determine, however, because for ri K0:95 AÊ (in VIII co-ordination), cations start to enter the M1 site in clinopyroxene. This means that unequivocal M2-liquid partitioning data cannot be obtained for cations such as Li‡ , Mg2‡ and Sc3‡ and hence that the D-ri curves cannot be as well-constrained as for plagioclase. Some indication of the relationships be1‡ 2‡ 3‡ tween EM2 , EM2 and EM2 may, however, be derived from observed dependencies of elastic moduli on charge in simple compounds and in cation polyhedra. Anderson and Anderson (1970) showed that, for an ionic crystal with electrostatic attractive forces and a

170

Fig. 4a,b Brice model ®ts to experimental data on clinopyroxene-melt partitioning: a clinopyroxene …Di91:6 En7:2 Jd1:2 † and silicate melt in the system diopside-albite (J.D. Blundy and J.A. Dalton, unpublished data); b clinopyroxene …Mg#cpx ˆ 0:86† and high-alumina basaltic melt (Hauri et al. 1994). Note that the dependence of partitioning on charge is similar to that for plagioclase (Fig. 3). Deviations of Pb2‡ and Mn2‡ from the best-®t curves in b may be related to their electronic structures (see text). In b the Mg content of the clinopyroxene M2-site …MgM2 † is calculated from the structural formula

Born power-law repulsive potential, the bulk modulus K is given by: Kˆ

AZa Zc e2 …n ÿ 1† 9do Vo

…11†

where Vo is the molecular volume, A is the Madelung Constant, Za and Zc are the anion and cation valences respectively, e is the charge on the electron, n is the Born power-law exponent and do is the interatomic separation. This equation demonstrates a linear dependence of bulk modulus on cation charge for ®xed structure type and molecular volume. Anderson and Anderson (1970) were able to go further and showed that a wide range of oxides with di€erent structure types obeys the empirical relationship: K ˆ 15:7…0:5†

Zo Zc e2 GPa Vo

where Zo is the charge on the oxygen anion. This relationship means that, in oxides, the bulk modulus is a linear function of cation charge and is inversely proportional to volume, or to the cube of the interatomic distance (in AÊ). Extension of these relationships to individual cationanion polyhedra was made by Hazen and Finger (1979) who found, for cations co-ordinated by oxygen in silicates and oxides, the following relationship (illustrated in Fig. 5):

Fig. 5 Plot of bulk modulus versus 3 cation charge …Zc † divided by metal-oxygen distance cubed …AÊ † for cation polyhedra in oxides (Hazen and Finger 1979) and for 1+, 2+ and 3+ ions in the M2 site of clinopyroxene (this study), and the cation site in plagioclase (Blundy and Wood 1994). Note that the linear correlation derived from compressibility data by Hazen and Finger (solid line) holds reasonably well in our results independently estimated from the ri -dependence of cation partition coecient (e.g. Figs. 3 and 4). Crosses denote the bulk moduli of the end-member minerals albite, anorthite and diopside

K ˆ 750…20†Zc d ÿ3 GPa

…12†

where d is the cation-anion distance, equivalent to ro plus the radius of the oxygen anion, rO2ÿ (ˆ 1:38 AÊ; Shannon 1976), and Zc is the cation charge. The bulk modulus K is therefore observed to be a linear function of cation charge in oxides and in oxygen polyhedra of the kind we are concerned with in clinopyroxene. To extend the observed correlation to Young's modulus, E, we need to consider the identity relating E and K: Eˆ

3K 1 ÿ 2r

171

where r is Poisson's ratio. Most minerals approximate Poisson solids and in this case, when r  0:25 we obtain: E  1:5 K

…13†

Therefore the value of ES in Eq. (10) should, like bulk modulus, be an approximately linear function of cation charge. This is exactly what we observed previously (Blundy and Wood 1994) for plagioclases, where 1+, 2+ and 3+ values of the Young's Modulus of the large cation site were found to be 67, 128.5 and 190 GPa respectively for anorthite and 52, 104 and 198 GPa for albite. When converted to site bulk modulus using Eq. (13) we ®nd, as shown in Fig. 5, that the elastic properties of the large cation site in plagioclase calculated from trace element partitioning are in very good agreement with the empirical relationship derived by Hazen and Finger (1979) from observed compressibilities of oxide polyhedra in minerals. Furthermore, the values obtained from trace element partitioning are, for 1+ and 2+ ions, in excellent agreement with the bulk properties of albite and anorthite respectively, suggesting that the bulk elastic properties are controlled, in this case, by the large cation site. Turning to Ca-pyroxene, we ®nd that the room temperature bulk modulus of pure diopside, 113 GPa (Sumino and Anderson 1984), when scaled to the size and charge of the M2 site, is also in good agreement with the Hazen and Finger relationship (Fig. 5). This suggests that one may be able to use Eq. (12) to estimate the 3‡ e€ective values of EM2 for the pyroxene M2 site. How3‡ ever, instead of forcing a value of EM2 on the pyroxene REE partitioning data we ®rst attempted to estimate 3‡ EM2 and ro directly in cases where there were sucient data. That is, we took experiments where four or more REE partition coecients were simultaneously measured (Grutzeck et al. 1974; McKay et al. 1986; Hart and Dunn 1993; Hauri et al. 1994; and J.D. Blundy, 6 unpublished points) and ®tted Eq. (10) to each experiment in turn using a general non-linear ®tting routine (Press et al. 1986, p. 526). The results of Hart and Dunn (1993) cover the entire range of REE radii and show that ro has a peak at about the radius of Ho, 1.015AÊ, with an esti3‡ mated value of EM2 of 280…50† GPa corresponding to a 3‡ KM2 value of about 187 GPa. The data of Hauri et al. (1994) (Fig. 4) show similar trends, with a plateau in D for REE with ionic radii between 0.977 and 1.027AÊ and K values of 145±215 GPa. In all, 16 sets of data were 3‡ treated in this way and the results for EM2 , converted 3‡ into KM2 , are plotted on Fig. 5. In Fig. 5, ®tted values of ro were converted to d, by adding 1.38AÊ for the radius of the oxygen anion. As can be seen, the results are scattered, but the average, about 180 GPa, lies quite close to the line of Hazen and Finger (1979). The reason for the 3‡ scatter is that ®t-values of EM2 and ro are highly correlated, so that, in those cases where ro is not very well constrained by the data, small errors in D value propa3‡ gate into large, correlated errors in ro and EM2 . 3‡ Because of the scatter in EM2 values we made no attempt to ®t the site modulus to an equation in temper-

ature, pressure or pyroxene composition. Instead, noting 3‡ that EM2 is close to what would be predicted from Eqs. (12) and (13) we constrained it to have a value corre3‡ sponding to KM2 of 180 GPa under P -T conditions close to the average used in the experiments i.e. at about 1:5 GPa and 1650 K. Under these conditions the di€erence in the bulk-crystal Young's Modulus of diopside and enstatite is only 12% relative (Anderson 1989, p. 3‡ 105). We therefore assumed that EM2 is independent of composition. We assumed further, from Eq. (13), that 3‡ the temperature and pressure derivatives of EM2 would bear a simple relationship to the temperature and pressure derivatives of the elastic properties of diopside. This follows from Eq. (12) and the observed relationship between the bulk modulus of diopside and the bulk modulus of the M2 site: 3‡ 2‡ EM2 ˆ 1:5EM2  1:5EDiop

Then, assuming that this relationship holds over all of pressure-temperature space, we obtain: 

3‡ @EM2 @T

and 

3‡ @EM2 @P



P

ˆ 1:5

  @EDiop @T P



  @EDiop ˆ 1:5 @P T T

Using the temperature and pressure derivatives of the bulk and shear moduli of diopside tabulated by An3‡ derson (1989) and constraining the average EM2 for experimental P -T conditions as above, we arrived at the following equation for 3+ cation substitution into the clinopyroxene M2 site: 3‡ EM2 ˆ 318:6 ‡ 6:9P ÿ 0:036T GPa

…14†

where P is in GPa and T in Kelvin. 3‡ Given this expression for EM2 we then used the nonlinear ®tting program to derive values of ro from a total of 82 experiments in which 3 or more pyroxene-liquid REE partition coecients were measured and in which the compositions of crystal and liquid phases were well constrained. These comprised 11 of the experiments discussed above, 25 experiments of Hack et al. (1994), 37 experiments described by Gallahan and Nielsen (1992) and a further 9 unpublished points of Blundy, which are described brie¯y in Blundy and Wood (1994). Adopted radii values for individual REE were those of Shannon (1976) for 3+ ions in VIII co-ordination. Resulting values of ro range from 0.979 to 1.055 AÊ. A similar range in optimum M2 site radius was obtained by Liu et al. (1992) who ascribed the variations to the e€ect of pressure. However, as noted by Liu et al., pressure is strongly correlated with crystal composition in clinopyroxene, particularly with Na and Al content. We performed stepwise linear regression of the derived values of ro against all major compositional parameters, pressure and temperature. The result is that the only important parameters are the Al content of the M1 site, and the Ca content of the M2 site. All other parameters,

172

including pressure, have F < 0:6 and cannot be considered signi®cant. We ®nd that the following equation ®ts the 82 points with a standard deviation of 0.009 AÊ (i.e. less than the di€erence in ionic radius between adjacent REE): M2 M1 Ê A ro ˆ 0:974 ‡ 0:067XCa ÿ 0:051XAl

…15†

M1 Ê is The derived dependence of ro on XAl …ÿ0:051A† consistent with crystal-chemical data on the change in M2 site radius on going from CaMgSi2 O6 to CaAl2 SiO6 (Smyth and Bish 1988) and with REE partitioning data in the system CaO-MgO-Al2 O3 -SiO2 (Blundy et al. 1996) which give ÿ0:04AÊ per Al atom in M1. For those experiments where there were 3 or more data points on 2+ ions (11 experiments) or on 1+ ions (9 experiments), we then used the ro values obtained from the REE 2‡ 1‡ partition coecients to derive EM2 and EM2 values using the ®tting program. These were converted to K values using Eq. (13) and are also shown in Fig. 5. Figure 5 demonstrates that the 1+, 2+ and 3+ ions obey the Hazen and Finger (1979) relationship approximately, but that the slope of the curve is slightly higher for pyroxene. Nevertheless, the general trend of the data is in accord with the suggestion that the e€ective E value for the clinopyroxene M2 site is a linear function of the charge on the substituting cation. A test of the applicability of Eqs. (14) and (15) to REE substitution in clinopyroxene is provided by their ability to predict the partition coecient of element G, Dg , from the value for element I, Di . Most of the studies in the literature involved measurement of one of the middle REE such as Sm and Gd. We used Eqs. (14), (15) 3‡ and (10) to predict ro , EM2 and D3‡ o for each of these experimental data then used the measured DSm or DGd to predict the partition coecients for the other REE and Y. These were compared with the measured values (251 total points) and are represented in Fig. 6 as:

Dcalc g Error …J † ˆ RT ln Dobs g

!

The standard deviation of 2762 J corresponds to an error in Dcalc of about 24%. Less than 45% of the exj perimental data were used to ®t Eq. (15) for ro , so this is a true test of predictive capability. Several observations should be made. Firstly, that the experimental data are overwhelmingly of electron-microprobe (EMP) origin and that the 1r uncertainty in Dj from EMP analysis is typically 20±50%. Therefore our equations predict the data to within experimental uncertainty. Secondly, the 1r error in La is typically of the order of 50% (Gallahan and Nielsen 1992) and of our 251 points, 72 are predictions for La which has increased the scatter in the results. (Note that for 47 of these from Gallahan and Nielsen (1992) Y was used as the predictor because Sm and Gd data were not available). Finally, 92% of the data lie between 4000 J in error corresponding to a predicted partition coecient of between 0.73 and 1.36 times the measured value. Since this is within the

Fig. 6 Histogram of 251 points showing the error in RT ln DREE which arises when the measured D for one middle REE is used, through Eqs. (10), (14) and (15), to calculate the other DREE values under the same conditions. Values on arrows denote equivalent fractional deviation from the true value at 1523 K. More than 92% of calculated DREE values are within the uncertainties in experimental measurement (see text)

uncertainty in EMP determinations of D, we consider that the e€ects of crystal composition and ionic radius on DREE for clinopyroxene are now fully constrained. In any experiment all that is now required is accurate measurement of one middle REE partition coecient and crystal bulk composition. Equations (10), (14) and 3‡ (15) can then be used to calculate ro , EM2 and D3‡ o and to predict the partition coecients for the other REE or for any other 3+ ion which enters the M2 site (e.g. Cm3‡ , Am3‡ ). Having established the validity of Eqs. (10), (14) and 3‡ (15) we can now calculate ro , EM2 and D3‡ for all o available REE partition coecient data. We performed this for 481 data points in the literature for which the partition coecients were measured using a microbeam technique, either EMP or ion microprobe. The data now form a matrix of points in pressure-temperature-composition space in which the e€ects of ionic radius and crystal composition have been removed. This means that, if one could correct for activity-composition relations of the melt phase, then, at ®xed pressure and temperature, all experiments, regardless of bulk composition and REE studied, should give the same value of Do . The next step is to attempt to correct for compositional e€ects in the melt phase.

173

A melt model Structurally, aluminosilicate melts are known to be complex, containing a variety of anionic species having only a resemblance to, rather than an equality with, the structures of the liquidus solid phases (e.g. Mysen 1990). Despite this complexity it appears that, over certain compositional ranges, silicate melts may behave thermodynamically as if they contained simple ordered structural units (Burnham 1981; Blundy et al. 1995). In the absence of complete speciation data, we attempted to take advantage of these latter observations as a means of systematising the dependence of partition coecients on silicate melt composition. Before considering REE, we use the same approach to investigate thermodynamically better constrained equilibria. Blundy et al. (1995) investigated the partitioning behaviour of sodium between clinopyroxene and silicate melts over a wide range of pressure and temperature. They showed that the crystal-liquid partition coecient DNa bears a very simple relationship to the equilibrium constant KNa for the melting reaction: NaAlSi2 O6 ˆ NaAlSi2 O6 crystal melt

…16†

Taking standard states of crystal and melt phase to be pure NaAlSi2 O6 crystal and liquid respectively at the P , T of interest then KNa is related to the standard state free energy change of the melting reaction, DGofusion : DGo ˆ ÿRT ln KNa ˆ ÿRT ln

amelt NaAlSi2 O6

!

acpx NaAlSi2 O6

…17†

Blundy et al. then converted activities to composition using the observation of Holland (1990) that acpx NaAlSi2 O6 at cpx M2 low XNaAlSi is approximately equal to X Na , the mole 2 O6 fraction of sodium on the M2 site in clinopyroxene, i.e. acpx NaAlSi2 O6 ˆ number of Na atoms per 6 oxygens

…18†

For the melt Blundy et al. (1995) used exactly the same assumption: amelt NaAlSi2 O6 ˆ number of Na atoms per 6 oxygens

…19†

This is equivalent to a quasicrystalline melt model (Burnham 1981) in which the melt is assumed to consist of pyroxene-like units. If this simpli®cation is a reasonable assumption, then DGo can be related to DNa , the crystal/liquid partition coecient, through Eq. (7). Given that the di€erence in mean molecular weight between crystal and liquid (generally less than 3% ) can be ignored, we obtain: DGo ˆ RT ln…DNa †

Fig. 7 The apparent 0:1 MPa free energy of fusion of jadeite, NaAlSi2 O6 , as a function of temperature. Solid line was obtained from the melting curve of pure jadeite (Blundy et al. 1995). Points were obtained from measured partitioning of NaAlSi2 O6 component between clinopyroxene and liquid in simple and complex systems and assuming ideal solution in both phases (see text). Agreement between data for pure jadeite and those for dilute NaAlSi2 O6 component assuming ideal solution con®rms that ideal behaviour (i.e. cpx cmelt NaAlSi2 O6 ˆ cNaAlSi2 O6  1:0† is a reasonable assumption

…20†

Although the melt model is a gross simpli®cation, Blundy et. al (1995) found that the standard state free energy change of the melting reaction derived from calorimetric and phase equilibrium data on pure NaAlSi2 O6 is in excellent agreement with the sodium partitioning data for a wide range of synthetic and

natural compositions (Fig. 7). The data shown in this ®gure span a wide composition range from the diopsidealbite join to natural tholeiite and picrite, over a pressure and temperature range of 0:1 MPa to 6 GPa and 1097±1800  C respectively. This means that the simple activity-composition relationship for NaAlSi2 O6 in melt (19) gives a reasonable representation of the partial molar free energy of the NaAlSi2 O6 component over most of the pressure, temperature and compositional ranges of importance in petrology. The only compositions which deviate from the simple activity-composition relationships are those containing substantial NaFe3‡ Si2 O6 component, and those on the diopsidealbite join where the SiO2 content is high enough for non-ideal CaMgSi2 O6 -SiO2 interactions to interfere (Blundy et al. 1995). In basaltic and picritic compositions, however, the simple melt and crystal models enable thermodynamic data on pure NaAlSi2 O6 to be used to calculate DNa within 17% for pressures of 1 GPa and above. This means that one point on the D-ri curve for 1+ cations can be determined accurately at any pressure and temperature. Then, from Eqs. (10), (14), (15) the remainder of the curve may be calculated for larger univalent ions such as K‡ , Rb‡ and Cs‡ , given a suit1‡ 2‡ able value for EM2 , e.g. 0:5 EM2 . Given the success of this model for Na pyroxenes, we assessed whether the same approach can be applied to the major Ca-components of igneous clinopyroxenes and, in turn to trace components containing REE and similar 3+ ions. Consider, therefore, the melting of diopside CaMgSi2 O6 : CaMgSi2 O6 ˆ CaMgSi2 O6 Diopside Melt

174

At the P , T of interest DGo for this reaction is related to the ratio of activities in melt and crystalline phases through an analogous equation to (17). Along the melting curve for pure diopside the equilibrium constant is 1.0 (both phases pure), hence: DGoPT ˆ DHTof ‡

ZT

0

B DCp dT ÿ T @DSTof ‡

Tf

ZP ‡

ZT Tf

1

DCp C dT A T

DV o dP ˆ 0

0:1

Or, rearranging: DHTof ÿ T DSTof ˆ ÿ

ZT

ZT DCp dT ‡ T

Tf

Tf

DCp dT ÿ T

ZP

DV o dP

…21†

0:1

where DCP , DS o and so on refer to the di€erences in thermodynamic properties between pure melt and pure crystal, P and T refer to the pressure and temperature of interest and Tf to the fusion temperature at 0:1 MPa …1665 K†. The left hand side of Eq. (21) is the free energy of fusion at 0:1 MPa with the heat capacity terms removed in order to linearise it with respect to temperature. Henceforth the term DHTof ÿ T DSTof will be referred to as the ``apparent 0:1 MPa free energy of fusion''. The integrals on the right hand side of Eq. (21) were evaluated along the diopside fusion curve of Williams and Kennedy (1969) to 5 GPa. Data for heat capacity for diopside crystals were taken from Robie et al. (1979) and for diopside melt from Lange and Carmichael (1990). Liquid volume data were obtained from Lange and Carmichael (1990) and crystal volume data as a function of temperature from Finger and Ohashi (1976). Bulk modulus data for diopside were taken from the summary of Anderson (1989) while liquid compressibilities came from Lange and Carmichael (1990). The pressure integral was evaluated from the Birch-Murnaghan equation using integration by parts (see Blundy et al. 1995). The terms on the right hand side of Eq. (21), calculated from the diopside fusion curve, are shown as a function of temperature (crosses) in Fig. 8. The slope of the curve should yield the DS o of fusion at Tf . The calorimetric value of DSTof , 82:5  1:2 J molÿ1 Kÿ1 (Richet and Bottinga 1986), is in excellent agreement with value obtained by regression, 83:4 J Kÿ1 at 1665 K, indicating that derived values of enthalpy, entropy and heat capacity changes of the fusion reaction to 5 GPa are all internally consistent. If good models of activity-composition relations for solid and liquid phases can be obtained, then, as previously shown for jadeite, calculated equilibrium constants for coexisting clinopyroxene and liquid in complex systems should be consistent with the results extracted from the diopside fusion curve. We took experimental data on basaltic and haplobasaltic liquids at high pressure and temperature from Walter and Presnall (1994), Kinzler and Grove (1992), and Blundy et al. (1995, appendix). These data apply to

Fig. 8 The apparent 0:1 MPa free energy of fusion of diopside, CaMgSi2 O6 , as a function of temperature. Solid line is the calorimetric measurements of Richet and Bottinga (1986), while crosses are values extracted from the melting curve of pure diopside measured by Williams and Kennedy (1969). All other points are from high pressure experiments on basaltic and related systems and were calculated by assuming ideal mixing of CaMgSi2 O6 component in liquid and solid cpx phases. Ideal mixing (i.e. cmelt CaMgSi2 O6 ˆ cCaMgSi2 O6  1:0† appears, over this composition range, to be a reasonable assumption

approximately basaltic liquids and span a wide P -T range (0:7±3:5 GPa; 1493±1873 K). For each experimental point we calculated the activity of CaMgSi2 O6 component in the crystal using the 2-site ideal model (Wood and Banno 1973): M2 M1 acpx CaMgSi2 O6 ˆ XCa
XMg

…22†

M2 M1 where XCa and XMg refer to atomic fractions of Ca and Mg on the clinopyroxene M2 and M1 sites, respectively, calculated using the method of Wood and Banno (1973). For the melt we used a quasicrystalline model in which the melt analysis is recast as the 6-oxygen components: NaAlSi2 O6 , KAlSi2 O6 , CaTiAl2 O6 , CaAl2 SiO6 , (Mg,Fe)3 Si1:5 O6 , Si3 O6 and Ca(Mg,Fe)Si2 O6 . Calculation of the components is straightforward (see Appendix) and provides the potential for CIPW-normative olivine, quartz and nepheline as well as hypersthene. We then obtained an estimate of CaMgSi2 O6 activity from the mole fraction of Ca(Mg,Fe)Si2 O6 component and the molar ratio of Mg to Mg ‡ Fe …Mg#melt †: melt acpx CaMgSi2 O6 ˆ XCa…Mg;Fe†Si2 O6
Mg#melt

…23†

This activity model is equivalent to that employed by Blundy et al. (1995) for NaAlSi2 O6 component in silicate melt. For each experimental point we calculated an apparent 0:1 MPa free energy of fusion from Eq. (21) with the activity terms on the right hand side to take account of mixing: DHTof ÿ T DSTof ˆ ÿ

ZT

ZT

DCp dT ‡ T Tf

ZP ÿ

Tf o

DCp dT T

DV dP ÿ RT ln 0:1

amelt CaMgSi2 O6 acpx CaMgSi2 O6

! …24†

175

Results, shown in Fig. 8, demonstrate that, as with NaAlSi2 O6 , the simple activity-composition relations produce good agreement between the fusion curve of pure diopside and the partitioning of CaMgSi2 O6 component between clinopyroxene and silicate melt for a wide range of bulk compositions. Calculated values only begin to deviate from the predicted curve under high P -T conditions (> 1800 K and > 3 GPa) where the normative olivine contents of the liquids are greater than 40%. Although the good agreement between predicted and observed partitioning implies that basaltic liquids saturated in clinopyroxene are structurally similar to diopside liquid, we emphasise that this is certainly not required by our observations. All we have shown is that, assuming basaltic liquids have pyroxene-like stoichiometry, then the activities of NaAlSi2 O6 and CaMgSi2 O6 components are roughly equal to their mole fractions. Importantly this provides a means of estimating DCa and DNa with good accuracy and hence ®xing one point on each of the 1+ and 2+ partition coecient curves (Fig. 2). With the ro Eq. (15) and the bulk modulusmolecular volume relationships of Fig. 5, they provide means of calculating the full D-ri curves for both 1+ and 2+ ions substituting on the M2 site.

Heterovalent substitution and Do for REE Having demonstrated that DNa and DCa may, with simple activity-composition relations, be related to the free energies of fusion of pure NaAlSi2 O6 and CaMgSi2 O6 pyroxenes, we now consider whether or not a similar approach can work for the REE. In this case the value of D3‡ o for a strain-free substituent, may be obtained from Eqs. (10), (14), and (15). The main diculty arises from the fact that no thermodynamic or melting data exist for end-member REE pyroxenes. In place of such data we need, as a minimum, a good understanding of the mechanism by which REE3‡ substitute for Ca2‡ on M2. There are several possibilities. The Ca substitution may be coupled to the replacement of a tetrahedral Si atom by Al, leading to a hypothetical REE pyroxene endmember REEMgAlSiO6 . Additional possibilities are replacement of 2Ca on M2 by Na ‡ REE, or of 3 Ca by 2REE plus a vacancy, leading to the hypothetical endmembers Na0:5 REE0:5 MgSi2 O6 and REE0:667 MgSi2 O6 respectively. In practice, all three substitutions can rigorously be described thermodynamically and are hence equally valid. Practically speaking, however, the experimental diculties involved in obtaining the vacancy concentration on M2 make the activity of the REE0:667 MgSi2 O6 component almost impossible to de®ne. We will therefore con®ne ourselves to the components Na0:5 REE0:5 MgSi2 O6 and REEMgAlSiO6 . Consider a melting reaction analogous to reaction (4), but involving the component Na0:5 REE0:5 MgSi2 O6 : Na0:5 REE:5 MgSi2 O6 ˆ Na0:5 REE0:5 MgSi2 O6 clinopyroxene melt

…25†

The enthalpy and entropy changes of this reaction at 0:1 MPa are related to the equilibrium constant (Eq. 24) as follows: DHTof ÿ T DSTof ˆ ÿ

ZT

ZT

DCp dT ‡ T Tf

ZP ÿ

Tf

DCp dT T

o

DV dP ÿ RT ln 0:1

amelt Na0:5 REE0:5 MgSi2 O6

!

acpx Na

0:5 REE0:5 MgSi2 O6

…26†

Let us assume that liquid and solid phases obey simple activity-composition relations similar to those exhibited by NaAlSi2 O6 and CaMgSi2 O6 components in liquid and crystalline phases. For crystalline pyroxene in the absence of short-range order on the M-sites we would anticipate activity-composition relations of the form: ÿ M2
0:5 ÿ M2
0:5 M1 acpx
XREE
XMg Na0:5 REE0:5 MgSi2 O6 ˆ XNa

while for the liquid, continuing the analogy with NaAlSi2 O6 and CaMgSi2 O6 , we might expect: 0:5 amelt
‰REEŠ0:5
Mg#melt Na0:5 REE0:5 MgSi2 O6 ˆ ‰NaŠ

where [Na] refers to the number of Na atoms and [REE] to the number of REE atoms per 6 oxygens. Applying Eq. (7) and assuming that liquid and solid activity coecients cancel we obtain: DHTof ÿ T DSTof ˆ ÿ

ZT Tf

ZP ÿ 0:1

ZT DCp dT ‡ T Tf

DCp dT T

! ÿ 3‡
0:5 M1 D0:5
XMg
fcrystalg Na
Do o DV dP ‡ RT ln Mg#melt
fmeltg

…27†

where D3‡ o refers to the ``strain-free'' partition coecient for a rare earth element with radius ro at the conditions of interest. Although the heat capacity di€erences between products and reactants are unknown, they can generally be assumed equal to zero over quite wide temperature ranges (e.g. Wood and Fraser 1976, p. 42). Once again the di€erence in mean atomic weight between coexisting natural liquid and clinopyroxene (0± 3% ) is negligible with respect to uncertainties in experimental values of DREE . Therefore (27) may be simpli®ed to: DHTof ÿ T DSTof ˆ ÿ

ZP

0:1

DV o dP ÿ RT ln

ÿ 3‡
0:5 M1 ! D0:5
XMg Na
Do …28† Mg#melt

The volume change of the reaction is also unknown, and certainly cannot, because of the large volume of fusion of jadeite, 22 cm3 molÿ1 at atmospheric pressure (Blundy et al. 1995), be assumed equal to zero. The volume of fusion of jadeite is, however an approximately linear function of pressure to 6 GPa, which means that the integral may be simpli®ed as follows: DHTof

ÿ

T DSTof

ÿ 3‡
0:5 M1 !   D0:5
XMg 1 oDV Na
Do 2 ‡ P DV ÿ P ˆ RT ln Mg#melt 2 oP …29†

176

We calculated DNa and D3‡ o for all data in the literature (427 points) for which sodium and REE partition coecients had been measured using a microbeam technique. Only data for which the melt was an anhydrous silicate were included. We then regressed the right hand side of Eq. (29) against T , P and P 2 to estimate the entropy, enthalpy and volume of fusion of the hypothetical Na0:5 REE0:5 MgSi2 O6 pyroxene. The data ®t Eq. (29) with a standard deviation of 2820 J and yield a value of DSTo of 80:16 J molÿ1 Kÿ1 and DV of fusion of 12 cm3 molÿ1 at 0:1 MPa. The results make thermodynamic sense in that the extrapolated volume of fusion of Na0:5 REE0:5 MgSi2 O6 pyroxene at 0:1 MPa …12 cm3 molÿ1 † is more than half the value for jadeite. The estimated entropy of fusion is also in good agreement with values for other pyroxenes: Richet and Bottinga (1986) give 82:5 J molÿ1 Kÿ1 for diopside while Blundy et al. (1995) estimate 64:8 J molÿ1 Kÿ1 for jadeite. Values of the right hand side of Eq. (29), corrected for pressure using the volume terms, were obtained for all the 427 available points of DREE measured by a microbeam technique. All points, for all REE, are plotted on the same diagram as a function of temperature in Fig. 9. The validity of our radius correction is clear, as is the temperature dependence and the simple activity-composition relations we have used. The points shown in Fig. 9 have a standard deviation …2820 J† which is essentially the same as that …2840 J† obtained by Blundy et al. (1995) for partitioning of Na between pyroxene and liquid. This means that our model ®ts the REE data as well as the Blundy et al. (1995) model ®ts the Na partitioning data. It should therefore be possible to use our ®t parameters, in conjunction with the Na partitioning equation of Blundy et al. (1995) to calculate

D3‡ o , and hence DREE for any rare earth element, from the crystal composition. In practice, however, we ®nd that, because of errors in measured Na contents of glass and pyroxene, a second equilibrium, that involving the REEMgAlSiO6 component, produces less scatter in calculated DREE . Consider the melting reaction: REEMgAlSiO6 ˆ REEMgAlSiO6 clinopyroxene melt

…30†

The equilibrium constant Ko is given by the activity ratio which must be related to measured concentrations in the two phases. For the clinopyroxene we follow Blundy et al. (1996) and the experimentally determined activitycomposition relations in aluminous clinopyroxenes (Wood 1987), with ideal solution on each individual sublattice, giving: M2 M1 acpx REEMgAlSiO6 ˆ XREE
XMg

…31†

Blundy et al. (1996) considered the e€ect of liquid bulk composition on measured REE partition coecients in the system CaO-MgO-Al2 O3 -SiO2 (CMAS) at 0:1 MPa. They showed that, after correcting for crystal composition, a 2-lattice model for the liquid, one lattice containing cations such as REE3‡ and Ca2‡ and the 4ÿ other anions such as Si3 O06 , AlSiO5ÿ 6 and Si2 O6 , could explain much of the dependence of partition coecient on liquid composition. We attempted to extend this model to the wide range of liquid compositions used in the experiments shown in Fig. 9, but obtained very scattered results. We therefore conclude that their model is not, at this stage, generalisable outside CMAS. We therefore returned to the simple model used for NaAlSi2 O6 , CaMgSi2 O6 and Na0:5 REE0:5 MgSi2 O6 melts: amelt REEMgAlSiO6 ˆ ‰REEŠ
Mg#melt

…32†

where [REE] refers to the atomic concentration of the REE in the melt on a 6-oxygen basis. These assumptions, again taking the ratio of mean atomic weights to be 1.0, lead to the following expression for equilibrium constant: DHTo

ÿ

T DSTo

!   M1 D3‡ 1 oDV o
XMg 2 P ˆ RT ln ‡ P DV ÿ Mg#melt 2 oP

…33†

where D3‡ o refers, as before, to the partition coecient of the ``strain-free'' REE substituent. We used 481 data points on all analysed REE to ®t Eq. (33) yielding, with a standard deviation of 3887 J, the following:

Fig. 9 The apparent 0:1 MPa free energy of fusion of hypothetical Na0:5 REE0:5 MgSi2 O6 pyroxene plotted as a function of temperature. The points are 427 simultaneously-measured Na and REE (or Y) partition coecients from the literature with the strain energy of REE substitution removed using Eq. (10). All REE elements are plotted on the same ®gure which demonstrates the validity of our radius correction and of simple ideal activity-composition relations for solid and melt phases

M1 D3‡ o
XMg RT ln Mg#melt

!

ÿ 7050P ‡ 770P 2 ˆ 88750 ÿ 65:644T

…34†

Again the results are thermodynamically reasonable. The estimated entropy of fusion …65:64 J molÿ1 Kÿ1 † is virtually identical to that of jadeite and the volume of fusion is about half that of the hypothetical Na0:5 REE0:5 MgSi2 O6 pyroxene and one third that of jadeite. This suggests that most of the volume of fusion

177

and 0±3% K2 O. We conclude that the simple thermodynamic models may be joined to the strain energy equations to produce an Eq. (34) capable of calculating REE clinopyroxene-liquid partition coecients with an uncertainty comparable to that involved in measuring them experimentally.

Practical application to D REE

Fig. 10 The apparent 0:1 MPa free energy of fusion of hypothetical REEMgAlSiO6 pyroxene plotted as a function of temperature. Available experimental data for all REE and Y are plotted. Again the strain energy of REE substitution was removed using Eq. (10)

of Na0:5 REE0:5 MgSi2 O6 is due to the Na-component. The ®ctive 0:1 MPa temperatures of fusion of the end-members are also in reasonable progression: 943 K for jadeite (Blundy et al. 1995), 1260 K for Na0:5 REE0:5 MgSi2 O6 , and 1352 K for REEMgAlSiO6 . Figure 10 shows the left hand side of Eq. (34) evaluated for all 481 points, plotted against temperature. As can be seen, the scatter is slightly worse than for the Na0:5 REE0:5 MgSi2 O6 component, but the temperature dependence is clear and, as suggested above, thermodynamically reasonable. Furthermore, the simple activity-composition relations appear to remove any observable melt-compositional dependence of partition coecient. Turning to the apparent standard deviation of 3887 J, one important contribution comes from the 27 unpublished points obtained on the join diopside-albite in our laboratory. We included them because they provide constraints on the ®t equation at pressures up to 6 GPa and at temperatures up to 2038 K. Several of the bulk liquid compositions are extremely sodic, however, containing up to 60 mol% NaAlSi2 O6 component and they show more scatter around the best-®t line than do most of the other data on more normal basaltic compositions. If we exclude these unpublished data from the estimate then the remaining 454 points ®t Eq. (34) with a standard error of 3543 J. Of this uncertainty, we have already shown that 2762 J accrues from the uncertainty in the strain-radius correction. Using normal error propagation this means that the errorsp introduced by the  thermodynamic approximations are …35422 ÿ 27622 † or 2217 J. This corresponds to a 1r error in calculated partition coecient, arising from the thermodynamic models, of 19%, a value smaller than that arising from the uncertainty in radius-strain correction. It should be noted that this relatively small scatter arises from a very wide range of bulk liquid compositions. The 454 points encompass anhydrous liquids with between 38 and 60 weight% SiO2 , 2±20% MgO, 0±22% FeO, 0±5% Na2 O

Having obtained a good description of partition coecient as a function of bulk composition, pressure and temperature, we now consider how the results may best be applied to petrological problems. Potential applications are very widespread and include the precise quanti®cation of DREE during polybaric fractional melting of the mantle, reconstruction of liquid REE compositions from cumulus and intercumulus clinopyroxene in layered intrusions, and quanti®cation of fractional crystallisation paths in systems crystallising clinopyroxene. The unsatisfactory conventional practice of adopting constant-value clinopyroxene-melt partition coecients from the literature when tackling complex polybaric, polythermal petrological problems involving clinopyroxene can now be discontinued. For a given petrological problem the accuracy of a calculated partition coecient depends on the nature of the available data. We will consider each of the possible scenarios in turn. In each case we will assume that pressure and temperature are approximately known. a) Crystal bulk composition and one REE partition coecient known This is the simplest and most accurate situation 1. Use Eq. (15) to calculate ro from the crystal composition. 3‡ 2. Use Eq. (14) to calculate EM2 at the pressure and temperature of interest. 3. Use Di , the known REE partition coecient, in conjunction with ri , the ionic radius of this element, to calculate D3‡ o from Eq. (10). 3‡ 4. With the values of D3‡ o , EM2 and ro the value of Di for any REE of radius ri is calculated from Eq. (10). This method has, as shown in Fig. 6, a greater than 90% probability of being within 0.73 times and 1.36 times the true value. b) Crystal and liquid bulk compositions known: no REE partitioning data available Repeat steps (1) and (2) as before. 3. Use the bulk compositions to calculate Mg#melt M1 and XMg in the clinopyroxene using the method of Wood and Banno (1973), i.e. Fe and Mg are partitioned equally between M1 and M2 sites.

178

Or

4. b If there is no coexisting olivine then the pyroxene composition must be used to estimate Mg#melt . This is more dicult because of the wide range of M2 contents of Fe and Mg which are present in natural clinopyroxenes. From the experimental data used in this study we cpx ®nd correlations between KFeÿMg and either temperature cpx or XMg . We used the latter to derive the following expression: cpx KFeÿMg ˆ 0:109 ‡ 0:186 Mg#cpx

…35†

where Mg#cpx is the molar ratio of Mg to Mg ‡ Fetotal in the clinopyroxene. The clinopyroxene composition is cpx therefore used to calculate KFeÿMg from (35) then, as in 4.a, the Mg#melt is obtained as follows: Mg#melt ˆ

cpx KFeÿMg Mg#cpx

cpx 1 ÿ Mg#cpx ‡ Mg#cpx KFeÿMg

We ®nd that this equation predicts the Mg#melt to within 18% in 90% of the 211 experiments used here. 5. Calculate D3‡ o from Eq. (34) 3‡ 6. With the values of D3‡ o , EM2 and ro the value of Di for any REE of radius ri is calculated from Eq. (10). d) Melt composition only known Fig. 11 Histogram of 454 points showing the error in RT ln DREE which arises when D3‡ o is obtained from the thermodynamic model of Eq. (34), and Eqs. (10), (14) and (15) are used to calculate the values of DREE under the same conditions. More than 92% of calculated DREE values are within 0.63 and 1.59 times those experimentally observed, as shown by the numbered arrows

4. Calculate D3‡ o from Eq. (34). 3‡ 5. With the values of D3‡ o , EM2 and ro the value of Di for any REE of radius ri is calculated from Eq. (10). As shown in Fig. 11 this method gives DREE with > 92% probability of being within 0.63 and 1.59 of the true value. c) Crystal composition only available Repeat steps (1) and (2) as before. M1 3. Calculate XMg for the clinopyroxene as in b) above. 4. a If coexisting olivine is present then Mg#melt can be predicted quite accurately from the forsterite content of the olivine, XFo :

Mg#melt ˆ

ol KFeÿMg XFo ol 1 ÿ XFo ‡ XFo KFeÿMg

ol where KFeÿMg is the ratio of Fe/Mg in olivine to that in ol the melt. Experimentally measured values of KFeÿMg ol are in the range 0.27±0.33 for most values of XMg and the error associated with the adoption of a value of 0.30 (Roeder and Emslie 1970) is only 5±10% for most bulk compositions. This error translates directly into the same percentage error in DREE .

This is the most dicult and probably the most common application. We approach this by using the liquid composition to calculate the composition of the coexisting pyroxene at the pressure and temperature of interest. This procedure is detailed in the Appendix. Despite the requirement to calculate the bulk composition of the pyroxene in order to determine ro and M1 XMg , this approach, calculating directly from liquid composition, produces surprisingly good results. Comparing calculated DREE with measured values we obtain a standard deviation of 3910 J (Fig. 12). Therefore in over 90% of cases, calculated D is in the range 0.60±1.67 of the true D, a scatter which is only about 8% worse than in the case where the pyroxene composition is actually known.

Concluding remarks We have shown that the partition coecients of REE and Y between clinopyroxene and liquid depend, as predicted from the Brice (1975) model, on the ionic radius of the cation, on the radius of the pyroxene M2 site into which they substitute, and on the apparent Young's modulus of the site. Using experimentally determined REE partition coecients obtained over wide ranges of pressure …0:0001±6 GPa†, temperature …1323±2038 K† and bulk composition we have shown the following: 1. The apparent Young's modulus of the M2 site depends, in agreement with the suggestion of Blundy and Wood (1994), approximately linearly on the charge on the cation such that:

179

4. Partitioning of the ``ideal'' REE (i.e. radius ro ) can be described thermodynamically in terms of equilibria involving either of the coupled substitutions Na ‡ REE , 2Ca or REE ‡ Al , Ca ‡ Si. A simple thermodynamic model for the melt in which 6-oxygen units such as CaMgSi2 O6 , NaAlSi2 O6 and REEMgAlSiO6 are assumed to mix ideally works well for anhydrous silicate melts over a wide range of composition. In practice we ®nd that the ideal partition coecient Do for element of radius ro may be calculated from: M1 D3‡ o
XMg RT ln Mg#melt

!

ÿ 7050P ‡ 770P 2 ˆ 88750 ÿ 65:644T

Combining this equation and the Brice Eq. (10) we ®nd that > 92% of REE partition coecients can be calculated to within 0.63±1.59 times the measured value. 5. Clinopyroxene bulk composition can also be calculated from the melt composition with sucient accuracy for ro to be estimated from Eq. (15). Only a small additional uncertainty arises in predicted DREE , relative to the case in which both clinopyroxene and melt compositions are known. Fig. 12 Histogram of 454 points showing the error in RT ln DREE which arises when the major element composition of the clinopyroxene is calculated from the melt composition (see Appendix). D3‡ o is then obtained from Eq. (34), and Eqs. (10), (14) and (15) are used to calculate the values of DREE under the same conditions. More than 90% of calculated DREE values are within 0.60 and 1.67 times those experimentally observed, as shown by the numbered arrows 1‡ 2‡ 3‡ EM2  12 EM2  13 EM2

Furthermore, the apparent Young's modulus for 2+ ions 2‡ substituting into M2, EM2 , is similar to the value for bulk diopside, indicating that the properties of the large M2 site control the bulk elastic properties of the crystal. This observation can be used to make approximations re3‡ garding the temperature and pressure sensitivity of EM2 : 3‡ EM2  1:5Ediopside ˆ 318:6 ‡ 6:9P ÿ 0:036T

GPa

2. The radius of the clinopyroxene M2 site depends primarily on its Ca content and on the Al content of M1: M2 M1 Ê ro ˆ 0:974 ‡ 0:067XCa ÿ 0:051XAl A

No other physical or chemical variables (including pressure) were found to be signi®cant. The total range in ro is 0:979±1:055AÊ (i.e. rLu ±rGd ), with an uncertainty of Ê The clinopyroxenes contain between 0 and 0:009A. 21% NaAlSi2 O6 component and have Mg numbers between 0.63 and 1.0. 3. Given the partition coecient of a middle REE such as Sm or Gd the values of D for all the other REE can be predicted accurately using the Brice Eq. (10) with uncertainties which are approximately the same as those involved in measuring D. In practice 92% of 251 values could be predicted to within 0.73±1.36 times the independent measurement.

We consider that the dependencies of clinopyroxene rare earth element partition coecients on pressure, temperature and bulk composition in anhydrous silicate melts are now suciently well known for the collection of experimental data on speci®c natural bulk compositions to be unnecessary. Experimental work now needs to be focused on understanding the details of meltcompositional e€ects which our simple melt model has lost in the scatter of the partition coecient data. In particular we require information on the e€ect of dissolved water on the partitioning of REE and other cations. At present the expressions introduced here remain untested on hydrous systems. Our model can be applied to a wide range of geochemical problems involving clinopyroxene. Not only does the model enable the appropriate DREE to be calculated for the pressure, temperature, composition conditions of interest, but for polybaric, polythermal processes variations in DREE in response to changing pressure and temperature can be quantitatively incorporated into models. Speci®c applications include: adiabatic decompression melting of the mantle, wherein at pressures less than  2:5 GPa clinopyroxene is the principal host for the REE; calculation of melt REE composition from clinopyroxene composition in cumulates or other plutonic rocks where melts are not preserved; prediction of the behaviour in magmatic systems of trace cations that are not amenable to experimental study, e.g. trivalent actinides. The approach is based on a simple causative link between mineral physics and trace element geochemistry, and is therefore readily generalisable to other mineral-melt systems. Acknowledgements B.J.W. acknowledges the support of the Alexander von Humboldt Stiftung at the Mineralogisches Institut,

180 UniversitaÈt Freiburg where most of this work was done. J.D.B. wishes to thank NERC and the Royal Society for research fellowships. A.W. Hofmann, C.J. AlleÁgre, W. van Westrenen and an anonymous reviewer provided informed comments on the manuscript and its unashamedly determinist philosophy.

Appendix: calculation of clinopyroxene composition and D REE from melt composition The methodology uses the liquid-solid partitioning of NaAlSi2 O6 and CaMgSi2 O6 components discussed above, to estimate the Ca and Na occupancies of the M2 site. Additional data required are Mg#, Ti, Cr and CaAl2 SiO6 contents of the pyroxene. Empirical equations for these were obtained from observed liquid-solid partitioning in the experiments discussed here and those of Falloon and Green (1987), Kinzler and Grove (1992) and Blundy et al. (1995; appendix). The method operates at ®xed pressure and iterates temperature, from an initial estimate, until the liquid reaches clinopyroxene saturation, which is taken to be the point at which the Ca ‡ Na content of the M2 site falls within a predetermined range. It works as follows: 1. Read in melt composition in weight% oxides (SiO2 , TiO2 , Al2 O3 , Cr2 O3 , FeO, MnO, MgO, CaO, Na2 O, K2 O); generally all Fe has been taken as FeO 2. Read in pressure (GPa) and an initial estimate of temperature (K) 3. Convert melt composition to moles 4. Calculate amounts of each cation (Si, Ti, Al, Cr, Fe, Mn, Mg, Ca, Na, K) to a total of 6 oxygens in the melt 5. Recalculate cations to 6-oxygen ``molecules'' as follows:

B ˆ 0:108 ‡ 0:323Mg#melt Mg#cpx ˆ

A A‡B

10. Calculate the activity of CaAl2 SiO6 component in the clinopyroxene …acpx CaTs †, from the product of Ca per 6 oxygens in the liquid and Al per 6 oxygens (less the Al in CaTiAl2 O6 †: acpx CaTs ˆ …CT ‡ CATS ‡ DI†  …CATS ‡ JD ‡ KT †   76469 ÿ 62:078T ‡ 12430P ÿ 870P 2  exp 8:314T Apply an upper limit to ensure pyroxene is not unreasonably aluminous: cpx IF …acpx cats > 0:4† THEN acats ˆ 0:4

11. Similar approach to calculate activity of CaMgSi2 O6 …acpx Di † from the …DI† content of the liquid using a parabolic pressure dependence of volume instead of a full Birch-Murnaghan treatment:   132650 ÿ 82:152T ‡ 13860P ÿ 1130P 2 acpx ˆ DI  Mg# exp melt di 8:314T cpx M2 12. Use calculated acpx CaTs and aDi to calculate XCa , the Ca content of the M2 site: M2 XCa ˆ

cpx acpx CaTs Mg#cpx ‡ aDi cpx cpx Mg#cpx …1 ÿ XCr ÿ XTi †

13. Check for clinopyroxene saturation by calculating whether Ca plus Na in M2 lies within the predetermined range, if not then modify temperature estimate

Na ˆ NaAlSi2 O6 …JD†

M2 SUM ˆ XCa ‡ Xjdcpx

K ˆ KAlSi2 O6 …KT †

TEST ˆ 0:92±0:55…1 ÿ Mg#cpx †

Ti ˆ CaTiAl2 O6 …CT †

IF…SUM > TEST† THEN T ˆ T ‡ 20

…Al ÿ 2Ti ÿ Na ÿ K† ˆ CaAl2 SiO6 …CATS †

RETURN to step …6†

Ca ÿ CaAl2 SiO6 ÿ CT ˆ Ca(Mg,Fe,Mn)Si2 O6 …DI †

TEST2 ˆ 0:85±1:1 …1 ÿ Mg#cpx †

1 3 ‰Mg

IF…SUM < TEST2† THEN T ˆ T ÿ 14

‡ Fe ‡ Mn ÿ Ca…Mg; Fe; Mn†Si2 O6 Š

ˆ …Mg,Fe,Mn†3 Si1:5 O6 …OL†

1 3 ‰Si

ÿ 2K ÿ 2Na ÿ 2Ca…Mg; Fe; Mn†Si2 O6 ÿ CaAl2 SiO6 ÿ1:5…Mg; Fe; Mn†3 Si1:5 O6 Š ˆ Si3 O6 …QZ †

6. Calculate the mole fraction of NaAlSi2 O6 , Xjdcpx , in the crystal from the expression of Blundy et al. (1995) using the jadeite content of the liquid …JD† at ®xed P (GPa) and T (K):   10367 ‡ 2100P ÿ 165P 2 Xjdcpx ˆ JD  exp : T  ÿ10:27 ‡ 0:358P ÿ 0:0184P 2 cpx 7. Calculate Ti content of clinopyroxene M1 site XTi using approximate relationship between CT in the liquid and Xjdcpx :   cpx ˆ CT  0:374 ‡ 1:5Xjdcpx XTi

8. Use similar empirical partitioning data to calculate Cr in cpx from moles of Cr in melt per 6 oxygens crystal M1 site XCr cpx XCr ˆ 5Cr

9. Calculate the Mg#cpx from Mg#melt (if there is no Fe in the system, set both values to 1.0 and by-pass this calculation) Aˆ

Mg#melt …1 ÿ Mg#melt †

RETURN to step …6† 14. Calculate the size of the M2 site  cpx  a M2 ÿ 0:051 CaTs ro ˆ 0:974 ‡ 0:067XCa M2 XCa 15. Calculate the equilibrium constant …KREE † for the reaction involving REEMgAlSiO6 component. (Note that the thermodynamic properties are very slightly di€erent from the quoted values which started from solid composition because we found a small average o€set of 241 J when starting from the liquid composition)   ÿ88509 ‡ 65:664T ÿ 7050P ‡ 770P 2 KREE ˆ exp 8:314T 16. Calculate D3‡ o from the equilibrium constant using crystal and liquid compositions   cpx  acats cpx cpx M1 ˆ Mg#cpx 1 ÿ XCr ÿ XTi ÿ XMg M2 XCa D3‡ o ˆ

Mg#melt M1 KREE
XMg

3‡ 17. Use Eq. (14) to calculate EM2 at the pressure and temperature of interest 3‡ 18. With the values of D3‡ o , EM2 and ro the value of Di for any REE of radius ri is calculated from Eq. (10)

181

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